This invention relates to multimode, circular-symmetric, isotropic, optical fiber waveguides and, more particularly, to optical fiber waveguides in which it is desirable to achieve minimum modal dispersion.
Optical fiber waveguides are now well recognized in the art as desirable mediums for transmitting optical information. Initially these optical fibers were constructed with a core having a uniform index of refraction surrounded by a cladding with a lower valued index of refraction. In this type of optical fiber waveguide with a stepped index profile and optical energy is coupled into the core, and the energy is transmitted to the far end of the optical fiber through a process of multiple reflections from the core-cladding interface.
One difficulty found with the multimode optical fiber having a stepped index profile is related to the fact that the various modes take widely different transit times. The modes which encounter very few reflections appear at the receiving end of the fiber much sooner than the modes which encounter many reflections. Since the latter modes are caused to travel through a longer length of the medium before reaching the receiving end of the fiber, this multimode effect causes any optical pulse transmitted through the fiber to encounter pulse dispersion.
A technique for reducing the effect of this multimode dispersion was disclosed in the article entitled "Multimode Theory of Gradient Core Fibers" by D. Gloge and E. A. J. Marcatili published in the November, 1973 issue of the Bell System Technical Journal pg. 1653-1678. In accordance with this technique, the index of refraction is caused to change along the radius of the fiber. The index of refraction at the core center has the highest value, and the index is changed in a roughly parabolic shape so as to decrease to the value of the index in the cladding at the core-cladding interface. The index profile in this type of fiber waveguide is given by the equation ##EQU4## where n.sub.1 is the on-axis refractive index, n.sub.2 is the refractive index of the cladding and of the fiber core at radius a, ##EQU5## and a is the core radius. For .alpha. = 2(1-.DELTA.) the fiber has an almost parabolic index profile, and the modes do not differ in transit time much from each other.
In the Gloge-Marcatili analysis, it was assumed that a parameter identified as profile dispersion is negligible. This parameter, profile dispersion, will be identified more completely hereinafter. At this point, it need not only be said that the profile dispersion is a function of the rate of change of the index with respect to wavelength.
The above-identified analysis by Gloge and Marcatili was extended in a very important way by D. B. Keck and R. Olshansky to optical fibers, wherein the profile dispersion is constant throughout the radius of the fiber core. See, for example, U.S. Pat. No. 3,904,268 entitled "Optical Waveguide Having Optimal Index Gradient" issued Sept. 9, 1975 to D. B. Keck and R. Olshansky. Where the profile dispersion is constant, it was determined by Keck and Olshansky that the index profile still follows a power law for minimum modal dispersion, but the exponent .alpha. in the index profile equation requires a value other than 2(1-.DELTA.). Specifically, the exponent .alpha. should satisfy the following equation for optimal modal dispersion: ##EQU6## where ##EQU7##
It has recently been determined though that the profile dispersion in graded index profiles is not constant with respect to radius for some of the dopants that are presently being used to shape the index profile. See, for example, the article entitled "Pulse Broadening in Multimode Optical Fiber With Large .DELTA.n/n" by J. A. Arnaud and J. W. Fleming to be published in Electronics Letters. In accordance with the mathematical technique presented by Arnaud and Fleming, the RMS impulse response width of an optical fiber can be determined using measured values of dn/d.lambda.. As pointed out in their article, the RMS impulse response width for fibers with germanium dopant do not achieve the optimum impulse response dictated by the theory established by Olshansky and Keck. As is further pointed out as in the Arnaud and Fleming article, this lack of agreement is due primarily to the fact that (1/n )dn/d.lambda. is not a constant as assumed in Olshansky and Keck's theory. In terms of the analysis to be presented hereinafter (1/n)dn/d.lambda. is assumed to be an arbitrary function of .lambda. and r, and consequently the results of Gloge et al and Olshansky et al are extended to cover a vast class of fibers.